1. Exact Results in Massive N=2 Theories
Konstantin Zarembo
(Nordita, Stockholm)
Integrability and Chaos in Multicomponent Systems, Vladivostok,

2. AdS/CFT correspondencez
5D bulk
strings
gauge fields
4D boundary

3. Breaking scale invariance
“IR cutoff”
horizon or
domain wall
massive field theory
asymptotically
AdS metric
approximate
scale invariance
at short distances
Routinely used in many contexts
Very few quantitative tests so far

4. Example: N=2* theory N=4 SYM hypermultiplets vector multiplet mass =

5. Holographic dual
Domain wall
“AdS6”
N=2* theory
AdS5
Pilch,Warner’00

6. Weak coupling Strong coupling UV regularization of pure N=2 SYM
In this talk:
N=∞
λ = gYM2 N
UV regularization of pure N=2 SYM
Weak coupling
Strong coupling
Hoyos’10
Dimensional crossover
by Eguchi-Kawai mechanism
Young,Z.’14

7. Holographic Wilson loops
Area law:
Maldacena’98
Rey,Yee’98

8. Example: Circle Minimal surface: Area: regularized area
Drukker,Gross,Ooguri’99
Berenstein,Corrado,Fischler,Maldacena’98
Area:
regularized area

9. Wilson loop in perturbation theory
Consider
- Gaussian vector field
- Gaussian scalar field
- circular contour

10. θ =

11. Summing rainbow diagrams
Random matrix model:
Large-N solution:
Wigner distribution

12. Strong-weak coupling interpolationλ
SYM perturbation
theory
String perturbation
theory 1 + +
+ …
Circular Wilson loop (exact):
Erickson,Semenoff,Zarembo’00
Drukker,Gross’00
Minimal area law in AdS5

13. Partition function of any N=2 theory on S4:
Localization
Pestun’07
Partition function of any N=2 theory on S4:
Vector multiplet:
Fund. hypermultiplet:
Adj. hypermultiplet:

14. Saddle-point equations:
Example: pure N=2
Saddle-point equations:
Decompactification limit
Douglas, Shenker’95

15. Super-QCD
Two phases:
Heavy quarks (M>μ)
Light quarks (M<μ)

16. N=2*

17. Weak Coupling OPE in condensates β-function of N=2 SYM!
Douglas,Shenker’95
OPE in condensates
β-function of N=2 SYM!

18. Phase transtion resonance on massless hyper:
Weak-coupling solution is valid up to
resonance on massless hyper:

21. Phase diagram
R: radius of S4

22. Strong coupling

23. Perimeter law (perimeter law) substitute <Φ> Buchel,Russo,Z.’13
Chen-Lin,Gordon,Z.’14
Z.’14

24. Dual geometry
Pilch,Warner’00

25. Minimal surface
for circle:
(domain wall region)

26. agrees with localization!
Minimal area
renormalized away
agrees with localization!

27. String fluctuations
Setup:
straight Wilson line

28. Semiclassical string path integral:

29. Fluctuation determinants

30. Density of states and phaseshifts
Cancels among
bosons and fermions

31. Divergences cancel among bosons and fermions
Area law
Lüscher
discrepancy?
vs. field-theory prediction:

32. Fradkin-Tseytlin term

33. Lüscher
Fradkin-Tseytlin
agree!
Matrix model:

34. Conclusions
Localization is a powerful probe of N=2 gauge theories
fisrt (?) quantum test of non-conformal holography
what are the implications of the phase transitions AdS/CFT?